Dear Colleagues,
This summer, the Department of Mathematics is running an online Special Colloquium Lecture Series on mathematical analysis and applications in allied fields. This lecture series is presented in conjunction with UTC’s NSF-funded Math REU Site. The first lecture will be given next Monday, June 27th by Dr. Chris Marx, Associate Professor of Mathematics at Oberlin College and Conservatory. I have included the details of Dr. Marx’s lecture below. Undergraduate and graduate students who are interested in Analysis, Partial Differential Equations, and Mathematical Physics are especially encouraged to attend.
Sincerely,
Roger Nichols
Speaker: Dr. Chris Marx (Oberlin College and Conservatory)
Date: Monday, June 27, 2022
Time: 2:00 p.m.-3:00 p.m. (EDT)
Zoom link: https://tennessee.zoom.us/j/94587985368
Zoom password: 591806
Title: Agmon-type Decay of Eigenfunctions for a Class of Schrödinger Operators with Non-compact Classically Allowed Region
Abstract: An important result by Agmon implies that an eigenfunction of a Schrödinger operator in $\mathbb{R}^d$ with an eigenvalue below the bottom of the essential spectrum decays exponentially if the associated classically allowed region is compact.
We extend this result to a class of Schrödinger operators with eigenvalues, for which the classically allowed region is not necessarily compactly supported: We show that integrability of the characteristic function of the classically allowed region with respect to an increasing weight function of bounded logarithmic derivative leads to $L^2$-decay of the eigenfunction with respect to the same weight.
Here, the decay is measured in the Agmon metric, which takes into account anisotropies of the potential. In particular, for a power law (or, respectively, exponential) weight, our main result implies that power law (or, respectively, exponential) decay of “the size of the classically allowed region” allows to conclude power law (or, respectively, exponential) decay, in the Agmon metric, of the eigenfunction.
The work presented in this talk is the result of joint research with my recent honors student, Hengrui Zhu (Oberlin ’21).